simple annuities final

8/8/2019 Simple Annuities Final

1/95

SIMPLE

ANNUITIESMATN02G

Group-01


2/95

SIMPLE ANNUITIES

Annuity

- sequence of equal payments made at equalintervals of time. The intervening time between

consecutive payments called the payment interval,the value of each payment is the periodic payment,denoted by R, whereas the period from the rstpayment to the last one is called the term of theannuity.

- in annuity that pays P3 every month for !years, the payment interval is 1 month, the termof annuity is 2 years and the periodic payment, R,is P300.


3/95

Two !assi"#ations o$ Annuity

a. "nnuity #ertain

- an annuity in which payments begin and end ona denite or $ed date. %nstallment payments forman annuity certain.

b. "nnuity &ncertain or #ontingent "nnuity

- is one whose payments e$tends over a periodof time the length of which cannot be foretoldaccurately. Pension and periodic life insurancepremium payments form an annuity certain.

SIMPLE ANNUITIES


4/95

Three %in&s o$ Annuity ertain

'. (rdinary "nnuity

- an annuity in which payments are made atthe end of each payment interval.

1 2 3 ' (0

P ))- P )) P )) P )) P ))

SIMPLE ANNUITIES


5/95


!. "nnuity )ue

- annuity in which payments are made at thebeginning of each period.

1 2 3 ' (0

P )) P )) P )) P )) -P ))

SIMPLE ANNUITIES


6/95


3. )eferred "nnuity

- annuity in which the last payment does notbegin immediately nor at the end of the rstperiod but at some later time.

- the length of time from the present to thebeginning of the rst payment interval is calledperio& o$ &e$erment*

1 2 3 ' (0

- P )) P )) P ))- -

SIMPLE ANNUITIES


7/95

AM+UNT AN, PESENT .ALUE+/ +,INA ANNUIT

"mount of (rdinary "nnuity *+

- sum of the compound amounts ofthe several payments each accumulated

at the end of the term.

P )) P )) P )) P )) P ))

P ))

P ))

P ))

P ))

P ))

S P ))


8/95

/+MULAS AM+UNT +/+,INA ANNUIT

S

n i

*'i n - '

i

+ R + R


9/95

Present /alue of (rdinary "nnuity *"

- sum of the present values of severalpayments, each discounted to the beginning ofthe term.

P )) P )) P )) P )) P ))

P ))

P ))

P ))P ))

P ))

P ))

AM+UNT AN, PESENT .ALUE+/ +,INA ANNUIT


10/95

A

n i

' - *'i -n

i

" R " R

/+MULAS PESENT .ALUE

+/ +,INA ANNUIT


11/95

EAMPLES

0$ample '1

2ind the amount and present value of an

annuity of P payable monthly for !years and 3 months if money is worth

'!4 compounded monthly.


12/95

5iven1

R P

i '!4 -------------------comp. monthly --- *'!46'! '4

n !.! years ---comp. monthly----*!.!7'! !8periods

+ 9 " 9

S

!8 .'

+

+ *3.:!:;

S P 1456(1*'6

*'.' !8 - '

.'

+

+ *3.:!:;

S P 1456(1*'6

EAMPLES


13/95

a

!8 .'

" R

" *!3.;


14/95

0$ample !1

=ou wish to save money on a regular basis

to nance an e$otic vacation in yeas. =ou are condent that, with sacrice and

discipline, you can force yourself todeposit P', annually at the end of

the ne$t years, into a savings accountpaying 84 annual interest nd what willbe the balance of your saving account

at the end of ve years.

EAMPLES


15/95

5iven1

R P',

% 84

n years+ 9

S

.8

+ '

+ '*.883;S P (57(0*7'

*'.8 - '

.'

+ '

+ ' *.883;

S P (57(0*7'

EAMPLES


16/95

0$ample 31

>r. (ng purchased a house. %f he paid

P!, as down payment andpromised to pay P!, every 3 months

for the ne$t ' years and the rate ofinterest is charged at '4 compounded

quarterly, what is the cash value of thehouse9

EAMPLES


17/95

5iven1

)own payment P !,

R P!,

i '4 -------------------comp. qtrly --- *'46? 3.84

n ' years ---comp. qtrly----*'7? ? periods#ash value down payment "

A

? .38

" !,

" !,*!.;:;;A P (15377*'7

' - *'.' -!8

.'" !

" !, *!.;:;;

A P (15377*'7

#ash value P !,

',388.?8 P 2(15 377*'7

EAMPLES


18/95

PEI+,I PAMENT +/+,INA ANNUIT

Periodic Payment of (rdinary "nnuity

- series of equal payment occurring

at the end of each period.


19/95

PEI+,I PAMENT +/+,INA ANNUIT

*'in - '

i

R +

' - *'i-n

i

R

"

+

S

n i

R

"

A

n i

R


20/95

T9E ANNUIT PAMENT/+MULA AN :E USE, /+;

"morti@ed loans

%ncome annuities

+tructured settlements >onthly house rent payments

>onthly installments

"nnual premiums of life insurance policy >onthly retirement benets *pension

plan


21/95

0$ample '1

%f money is worth :4 compoundedquarterly, how much must be

deposited every 3 months in a fund inorder to have P', at the end of

' years9

EAMPLES


22/95

5iven1

+ ',

i :4 -------------------comp. qtrly --- *:46? !4

n ' years ---comp. qtrly----*'7?


23/95

0$ample !1

%f an investment has a present value of

P':3,, what is the periodicpayment required if the interest rateis ?4 within < years and ; months9

EAMPLES


24/95

5iven1

" ':3,

i ?4

n


25/95

(n Aune ', !'3, 5randma deposits P',833into an account to help pay for 0milyBs

summer volleyball camp for four

consecutive years. The rst camp isscheduled for Aune !'?. The account earnsay 3' ofeach year. %f 5randma wants the balance to

be P at the end of the four years, howmuch should she withdraw for 0mily each

Aune9

EAMPLES


26/95

5iven1

" ',833

i


27/95

/IN,ING INTEEST ATE

%f the interest rate is to bedetermined, we use linear

interpolation and by formula.


28/95

LINEA INTEP+LATI+N

- %t is a method of curve ttingusing linear polynomials.

- %t calculates the unDnown rate asif it lies on a straight line between thetwo rates.

- %t is the simplest way to calculatethe unDnown rate.


29/95

5%/0E1

R P :

+ P 38,

m ? *qtrlyn 8F years ----------comp. qtrly --- *8.!7? !; periods

S

n i

+ R

S

!; i38, :

S

!; i 38,

:

'4*87(

EAMPLES


30/95

'6! 4 d

34 ?.!'::

$ ?

G 4

'.3.;

d '.3.;

$ G

d .!??' 4$ small i d

3. .!??' 4 3. !??' 4

i $ 7 m

3.!??' 4 $ ?

12*68 <

EAMPLES


31/95

"n item can be bought for P , orfor P , down payment and P ',; amonth for !? months. 2ind the ratecompounded monthly.

Gi=en ;

#ash price P ,

n !? mos.

m '! *compounded monthly)p P ,

R P ',;

EAMPLES


32/95

So!ution" P , H P ,

P ?,

A

n i" R

A

!? i?, ',;

A

!? i ?,

',;

23*0746

EAMPLES


33/95

'6!? 4 d

86!?4 !3.'?

d .!?< 4$ small i d

.!;'8 .!?


34/95

>0TI()+

Tableof

"nnuity

2ormul

a

/IN,ING T9E TEM +/ANNUIT


35/95

2uture /alue

•

Present /alue

•

S"mount ofannuity

A Present value ofannuity

Periodicpayment

n Term or period

i %nterest rate

TA:LE +/ ANNUIT


36/95

2&T&R0 /"J&0PR0+0ET

/"J&0

(rdinary "nnuity

"nnuity )ue

/+MULA


37/95

Iow many payments would be needed ifthey oKered P';,


38/95


39/95

EAMPLES

n - ln *'-

';,


40/95

Iow long does he need to accumulate a sum ofP',, if a man deposits P at the end of

each quarter in a fund which earns ?4compounded quarterly.

EAMPLES

s

n i+ R

s

n i +

R

s

n .?',

s

n .? 20

Jocate at the table of 2/("


41/95


42/95

+NLU,ING PAMENT +IGINALAM+UNT /ULL PAMENT

+': *';.


43/95

EAMPLES

n ln *'

'

.'

ln *' .'

n ln*'.!ln*'.'

n .':!3!.;;

n 18*323 perio&s


44/95

+ P',!:;.!:

i' 4 annually or 4semiannually

E 9

R P semiannually

EAMPLES

s

n i

+ R

s

n i +

R

s

n .',!:;.!

:s

n . 20*(76Jocate at the table of 2/")


45/95


46/95

EAMPLES

n ln *

',!:;.!: $. * '.

'

ln *'

.

n ln *'.;8;;

ln *'.

n .


47/95

" P3,'!8.?':

i < 4 annually

E 9

R P


48/95


49/95

EAMPLES

n - ln L '.<* ' -

3'!8.?':


50/95

The Present value of an annuity due *" isthe sum of the present values of the paymentsat the date of the rst payment.

Present /alue of the annuity due is its rstpayment at the beginning of the term plus thepresent value of the ordinary annuity of n-'payments.

The ">(&ET *+ of an annuity due is the

sum of all accumulated amounts of thepayments at the end of the term.

AM+UNT AN, PESENT.ALUE +/ ANNUIT ,UE


51/95

#lues to help identify annuities due1

NPayments at the beginning of

eachO NPaymentsO in advance

N2irst paymentO made today

NPaymentsO starting now



52/95

Present /alue formula1


an-' i" R R

' - *'i -n

i" R *'i


53/95

" lottery oKers the winner a choicebetween a P3, cash pri@e, or

quarterly payments of P8, beginningimmediately and continuing for ! years.Qhich alternative should the winner picD

if money is worth :4 compounded

quarterly9

EAMPLES


54/95

+olution1

EAMPLES

a:-' .!

" 8, 8,

' -

*'.!-:

.!" 8, *'.!

A 283577(*83

" 8, *?.3;?3:


55/95

The monthly rental of an apartment in>alate is P;, payable at the

beginning of each month. %f money isworth '!4 compounded monthly, whatis the cash equivalent of one*' years

rent9

EAMPLES


56/95

+olution1

EAMPLES

a

'!-' .'

" ;, ;,

' - *'.'-

'!.'" ;, *'.'

A 1025308*4(

" ;, *''.3


57/95

2uture /alue formula1

The 2uture /alue of annuity due formula is

used to calculate the ending value of a seriesof payments or cash Sows where the rstpayment is received immediately.


sn' i

+ R - '*'in - '

i+ R *'i


58/95

Iow much will 0lyse accumulate in herRR+P by age


59/95

+olution1

EAMPLES

s

n' i

+ R - '*'in - '

i

+ R *'i

*'.?


60/95

+uppose that an individual would liDe tocalculate their future balance after

years with today being the rst deposit.

The amount deposited per year is P',and the account has an eKective rate of34 per year. %t is important to note that

the last cash Sow is received one yearprior to the end of the th year.

EAMPLES


61/95

+olution1

EAMPLES

s

n' i+ R - '

*'in - '

i+ R *'i

*'.3 - '

.3

+ ', *'.3

+ ', *.?


62/95

PeriodicPaymentwhen " isDnown

PeriodicPayment when +is Dnown

PEI+,I PAMENT +/ANNUIT ,UE


63/95

" company wishes to deposit an amount ofmoney into an account at the beginning of

each year for the ne$t years to purchasea new machine costing . Iowmuch will each yearly payment be if the

current interest rate is 8.!49

EAMPLES


64/95

/.A, > (05000

1 2 3 ' (0

N ( years r 7*2< peryear

?? ?? ?? ?? ??

+(J&T%(E:

EAMPLES


65/95

EAMPLES

R

*'.8!*'.8! -' .8!

,

R

*'.8!.?'8:8:?! .8!

,

R


66/95

"n individual who would liDe to calculatethe amount they can withdraw once per

year in order to allow their savings to last years. +uppose their current balance,

which would be the present value, is, and the eKective rate on the

savings account is 34.

EAMPLES


67/95

P.A, >(5000

1 2 3 ' (0

N ( years r 3< per year

?? ?? ?? ?? ??

+(J&T%(E1

EAMPLES


68/95

EAMPLES

R

'' - *'iH*n-'

i

"

R

'' - *'.3H*-'

.3

,

R ?.8'8;:?3

,

150(6*67

PESENT .ALUE +/ A


69/95

PESENT .ALUE +/ A,E/EE, ANNUIT

)eferred annuities are annuities that donot maDe payments until later.

The present value of a deferred annuity isthe value at the beginning of the periodof deferment, not at the beginning of

the term of the ordinary annuity.

PESENT .ALUE +/ A


70/95

PESENT .ALUE +/ A,E/EE, ANNUIT

Adn i

" R -A

id

i

" '-*'i H*nd '-*'i H*nd

-

i

AM+UNT +/ A ,E/EE,


71/95

S

n i+ R

AM+UNT +/ A ,E/EE,ANNUIT

*'i n - '

i+ R


72/95

2ind the present value of deferred annuityof P', a year for ' years deferred for

years. >oney is worth 8.4 eKective.

5iven1

R ',

% 8.4

d years

n ' years

EAMPLES


73/95

A'

8." ',

- A8.

.8

" ',

'-*'.8 H*'

'-*'.8 -

-

.8

" ', *:.:!8'! - ?.?::

" '3!,?


74/95

%f money is worth '


75/95

EAMPLES

A'!<

:" 3, -

A

:<

.:

" 3,'-*'.: H*': '-*'.: -<

-

.:

" 3, *;.38':; H ?.

" !:,''.


76/95

PEI+,I PAMENT,E/EE, ANNUITIES

Perio&i# Payment ,e$erre& Annuities;

"n annuity contract can allow an

investor to maDe periodic payments on ascheduled basis, either monthly, quarterlyor annually. The annuity pays out itsbenets at a later date and is referred to

as a periodic-payment deferred annuity.

PEI+,I PAMENT @ +/


77/95

PEI+,I PAMENT @ +/,E/EE, ANNUIT

To obtain the P0R%()%# P"=>0ET *R usethe formula1

"and i

R

-a

d i

i

R

'-*'i H*nd

'-*'i Hd-

i

"


78/95

Qhat equal payment at the end of eachmonth for ? years will pay oK a debt of

':,, the rst payment is due in ' yearand ? months9 >oney is worth !?4compounded monthly.

EAMPLES


79/95

5iven1" ':,

U !?4

m '!

n ?*'! ?: *no. of payment to bemade

d *' ?6'!*'!-'

*' '63*'!-'' *no. of ctitious

payments

EAMPLES


80/95

EAMPLES

':,a

?:' .!

R

-a

; .!

':,a


81/95

EAMPLES

.!

R

'-*'.!H*?:'

'-*'.! H'-

.!

':,

R

3.


82/95

"n heir inherits !,. %nstead of taDingthe cash, she invests the money at


83/95

d ; *ctitious payments

n ! *no. of actual payments to bemade

" !, U


84/95

EAMPLES

!,a

!;

.3

R

-a

; .3

!,a

!; .3R

-a

; .3

175('0*2(

R ';.'::? H 8.8:


85/95

EAMPLES

.3

R

'-*'.3 H*!;

'-*'.3 H;-

.3

!,

R

';.'::? H 8.8:


86/95

SUMMA +/ /+MULAS

S

n i+ R

*'i n - '

i+ R

a

n i

' - *'i -n

i

" R

" R

"mount1 Present /alue1

(R)%E"R= "EE&%T=1


87/95

SUMMA +/ /+MULAS


P0R%()%# P"=>0ET (2 (R)%E"R= "EE&%T=1

+

S

n i

R

*'in - '

i

R +

"

a

n i

R

' - *'i-n

i

R "


88/95

SUMMA +/ /+MULAS

ThrouBh !inear interpo!ation

2%E)%E5 TI0 %ET0R0+T R"T0 (2 "EE&%T=1


89/95

SUMMA +/ /+MULAS

"mount - (rdinary Present /alue -(rdinary

2%E)%E5 TI0 T0R> (2 "E "EE&%T=

n ln *'

+

R i

ln *' i

n - ln *' -"R i

ln *' i

n ln *

+ $iR *'

i

'

ln *' i

n - ln L 'i * '-

"

R '

ln *' i

M

"mount H "nnuity)ue Present /alue H "nnuity)ue


90/95

SUMMA +/ /+MULAS


"EE&%T= )&0

a

n-' i" R R

' - *'i -n

i" R *'i

sn' i

+ R - '

*'in - '

i+ R *'i


91/95

SUMMA +/ /+MULAS


P0R%()%# P"=>0ET (2 "EE&%T= )&01

R

'

' - *'iH*n-'

i

"R

*'i*'i n -'

i

+

+

S

n i

R "an-'

i

R

*'i

'


92/95

SUMMA +/ /+MULAS

S

n i+ R

*'i n - '

i+ R

"mount1

)020RR0) "EE&%T=1


93/95

SUMMA +/ /+MULAS

Present /alue1

)020RR0) "EE&%T=1

a

nd i" R -

a

id

i

" '-*'i H

*nd

'-*'i Hd-

i


94/95

SUMMA +/ /+MULAS

P0R%()%# P"=>0ET (2 )020RR0) "EE&%T=1

+

S

n i

R

*'in - '

i

R

+

"mount1


95/95

SUMMA +/ /+MULAS

Present /alue1

P0R%()%# P"=>0ET (2 )020RR0) "EE&%T=1

"a

nd i

R

-a

d i

R

'-*'i H '-*'i Hd

"

simple annuities final

Documents